## Verifying temporal properties of systems with applications to petri nets

##### Abstract

This thesis provides a powerful general-purpose proof technique for the verification
of systems, whether finite or infinite. It extends the idea of finite local
model-checking, which was introduced by Stirling and Walker: rather than
traversing the entire state space of a model, as is done for model-checking in
the sense of Emerson, Clarke et al. (checking whether a (finite) model satisfies
a formula), local model-checking asks whether a particular state satisfies a formula,
and only explores the nearby states far enough to answer that question.
The technique used was a tableau method, constructing a tableau according to
the formula and the local structure of the model. This tableau technique is here
generalized to the infinite case by considering sets of states, rather than single
states; because the logic used, the propositional modal mu-calculus, separates
simple modal and boolean connectives from powerful fix-point operators (which
make the logic more expressive than many other temporal logics), it is possible
to give a relatively straightforward set of rules for constructing a tableau. Much
of the subtlety is removed from the tableau itself, and put into a relation on the
state space defined by the tableau-the success of the tableau then depends on
the well-foundedness of this relation.
This development occupies the second and third chapters: the second considers
the modal mu-calculus, and explains its power, while the third develops
the tableau technique itself
The generalized tableau technique is exhibited on Petri nets, and various
standard notions from net theory are shown to play a part in the use of the
technique on nets-in particular, the invariant calculus has a major role.
The requirement for a finite presentation of tableaux for infinite systems
raises the question of the expressive power of the mu-calculus. This is studied in
some detail, and it is shown that on reasonably powerful models of computation,
such as Petri nets, the mu-calculus can express properties that are not merely
undecidable, but not even arithmetical.
The concluding chapter discusses some of the many questions still to be
answered, such as the incorporation of formal reasoning within the tableau
system, and the power required of such reasoning.